We investigate the ability of current cosmic microwave background data to reliably constrain the form of the primordial power spectrum generated during inflation. We attempt to identify more exotic power spectra that yield equally good fits to the data as simple power-law spectra. In order to test a wide variety of spectral shapes, we combine the flow formalism, which is a method of stochastic model generation, with a numerical integration of the mode equations of quantum fluctuations. This allows us to handle inflation models that yield spectra that are not well described by the standard spectral parameterization. Using the latest WMAP data set, we find a high degree of variation in possible spectral shapes. In particular, we find strongly running spectra arising from fast-rolling inflaton fields providing equally good fits to the data as power-law spectra arising from slowly rolling fields. Current data poorly constrains the spectrum on scales k<0.01h Mpc⁻¹, where the error due to cosmic variance is large. Among the statistically degenerate models, we identify spectra with strong running on these larger scales, but with reduced running at smaller scales. These models predict values for the tensor-to-scalar ratio, r, that lie outside the 2-σ confidence interval obtained from SDSS + WMAP (Sloan Digital Sky Survey plus Wilkinson Microwave Anisotropy Probe) data for spectra that are parameterized as power laws or spectra with constant running. By considering more generalized power spectra, we therefore open up regions of parameter space excluded for simpler models.